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Operations on Maclaurin Series
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Operations on Maclaurin series including term-by-term differentiation and integration, adding and subtracting coefficients, multiplying and dividing series to find initial terms, using substitutions for composite functions, and applying series expansions to evaluate limits.
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Maclaurin Series of Composite Functions
AHL 5.19
The Maclaurin series for a composite function f(g(x)) is
f(g(x))=f(0)+g(x)f′(0)+2![g(x)]2f′′(0)🚫
Basically, we replace xn with [g(x)]n.
Adding and Subtracting Series
AHL 5.19
We can add or subtract Maclaurin polynomials by adding or subtracting the coefficients of each power of x from the original polynomials.
Integration of Maclaurin Series
AHL 5.19
We can integrate a function by integrating its Maclaurin polynomial term by term.
Differentiation of Maclaurin Series
AHL 5.19
We can differentiate a Maclaurin series using the power rule on each term.
Multiplying Maclaurin Series
AHL 5.19
We can find the first few terms in the Maclaurin Series for a product f(x)×g(x) by multiplying out the first few terms in the Maclaurin Series for f and g.
Division of Maclaurin Series
AHL 5.19
We can perform "division" of Maclaurin series by assuming that the result of the division is a series with
p(x)=g(x)f(x)=a0+a1x+a2x2+⋯,
where an are the coefficients of the nth power term. Then, we multiply g(x)f(x) and a0+a1x+a2x2+... by g(x). Finally, we solve for the unknown coefficients using the Maclaurin series of f(x).
Limits using Maclaurin Series
AHL 5.19
Certain limits may be evaluated with L’Hôpital's rule but require difficult or repeated differentiation. Sometimes, expanding a function within the limit using its Maclaurin series and simplifying is an easier method for evaluating these limits.
This kind of simplification often works because we know that after a certain point, every term will tend to 0, allowing us to focus on the few terms that do not.
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